#110930

order	512
nilpotency class	3
isoclinism	isoclinic class of #110920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

782 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,15,20,30)(12,22)(13,25)(14,17)(16,24)(18,27)(19,33,35,36)(21,32,26,29)(23,37)(28,39,40,34)(31,38);(1,2,4,7,5,8,3,6)(9,11)(10,15,17,30)(12,19)(13,26)(14,20)(16,28,31,40)(18,32,25,29)(21,27)(22,36,37,33)(23,35)(24,39)(34,38);(9,12,13,24,14,23,27,38)(10,16,18,22,17,31,25,37)(11,19,26,39,20,35,21,34)(15,28,32,36,30,40,29,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18126);